MAKING SENSE OF THE EQUATION SHEET
Interference & Diffraction
INTERFERENCE
    d sin  . Equation for path length difference. r1 – r2 is completely general. Use  sin  only when the two sources are far away from the observation point.



is completely general whenever you have waves from two sources interfering.
2  
 
  d sin


 d 

 d y

applies to interference from multiple slits.  is the phase difference between
2 L waves from successive slits at the point of observation. d is the slit separation.  is the wavelength.  is the position on the screen measured as an angle. y is the position on the screen measured as a distance. L is the distance from the slits to the screen. sin    that: sin n d

I  4 cos 2
applies to interference from multiple slits.
  successive peaks.
  n d

for small angles and that
 is the angular position of the n

 d
where  
applies only to the superposition of 2 waves. th order peak. Note
is the angular separation between

DIFFRACTION a
 a sin  applies to diffraction.  a
is the path length difference between the top and bottom of the slit of width a .

2 
 ...
applies to diffraction. Here bottom of the slit.
 is the phase difference between the waves coming from the top and the sin    m a

applies to diffraction.  is the angular position of the m th order minimum caused by diffraction.

INTERFERENCE PLUS DIFFRACTION
I
1
I
 sin
  / 2

 / 2
 2
gives the shape of the diffraction pattern (the envelope).
I
N
I
 sin

N  / 2 sin
  / 2

  2

gives the shape of the interference pattern (the peaks). N is the number of slits.
Note that: I I

 sin
  / 2

 / 2

  sin


N  / 2 sin  / 2

  2


gives the total intensity pattern.
RESOLUTION OF LENSES, GRATINGS, ETC
 
 a
is the minimum angular separation of two objects resolvable through a 1D slit of width a.

0
 1.22

D is the minimum angular separation of two objects resolvable through a lens or circular aperture of diameter D.  c
can also be taken to mean the minimum resolvable angle.
 
 min 
1
applies to resolution of two interference peaks through a diffraction grating.  
Nm resolvable wavelength difference. N is the number of slits. m is the order of the peak.
is the minimum

MAKING SENSE OF THE EQUATION SHEET
Quantum Physics, Part I
ENERGY & MOMENTUM
KE max
 eV stop
 hf     off the metal is KE max
. V stop the metal.
 f
0

applies to the photoelectric effect. The maximum kinetic electrons coming
is the stopping voltage. hf is the energy of the photon.  is the work function of
Note: Multiplying any voltage V by electric charge e gives energy in eV numerically equal to the voltage.
For example: If V  69 volts, then

69 volts
  69 eV .
KE 
1
2 mv 2  p 2
2 m
gives the kinetic energy for any massive particle. Note that a photon is not a massive particle.
E  pc hf  hc

 wavelength.

gives the energy of a photon. For
  applies to both massive particles and photons.

use nanometers for
KE p 2
  h
2 m 2 m
2
 2
gives the kinetic energy of any massive particle.
SCHRODINGERS EQUATION
  i

 t
KE 
 2
2
is for electrons.
is the time dependent schrodinger equation. Here capital psi  is a function of 
2  x and t.
  
2


    is the time dependent solution to the schrodinger equation. Lowercase psi   
is solution to the time independent schrodinger equation.

2 
2 
2 
2
   E  is the time independent schrodinger equation. E is the energy of the particle.
 *
        2
is the probability density function, it gives the probability per unit length that the particle can be found at x. The * denotes complex conjugate.

P ab
  a b    2 dx
    Ae
gives the probability that the particle can be found between x=a and x=b.
   t

is the solution to the schrodinger equation for a free particle (the potential energy U(x) is zero).
Note that    E , the energy of the particle.
2 m
is the Heisenberg uncertainty principle. The uncertainty in momentum multiplied by the uncertainty in position must be greater than or equal to .
 n

2
L sin n 
L x gives the n th state wavefunction for a particle in an infinite square well of length L.
E n

2
2 
 n  

2


 h 2
8 mL 2

 n 2  E n 2 gives the n th state energy for a particle in an infinite square well of length L. n   2 L gives the n th state wavelength of the wavefunction for a particle in an infinite square well of length L.

MAKING SENSE OF THE EQUATION SHEET
Quantum Physics, Part II
THE FINAL STUFF
T e  2 KL where K 2 
2 m

U o
 E

2
T is the probability that a particle of energy E can tunnel through a potential energy barrier of length L and height
U
0
. t
0

2 h

E
2
 E
1

This equation gives the half-period of the time-dependent wavefunction that results from a superposition of two stationary states.
 
 
 e 2 r
The “coulomb potential”, or in other words, the potential that an electron in a hydrogen atom “feels”. e is the electric charge (and here we assume there is a single proton; otherwise it would be e(Ze)). r is the distance to the nucleus.   1/ 4 
0 is a constant.
   
8 abc sin

 n
1 a
 x
  n
2 b
 y
  n
3 c
 z

This is the wavefunction for a particle in an 3-dimensional infinite square well of lengths a, b, c, in the x, y, z directions respectively. n1, n2, and n3 are independent of each other, but must be >1.

3


8 h 2 
 n 2
1  n 2
2  m a 2 b 2 c n 2
3
2


Allowed energies for the particle in 3D infinite square well.

1 s
 r   

1 e

0
 a
0
3
The ground state wavefunction for the electron in the hydrogen atom. a
0 is the bohr radius.
E n

 13.6 eV n 2
: Energy levels for the hydrogen atom.

E n
  13.6 eV
Z n 2
2
Energy levels for an electron subject to Z positive charges. Note that Z=1 gives the hydrogen equation.
 4 r
 
2 dr
You probably won’t have to use this equation. What it means it that probability per unit of radial distance is equal to
 nlm
 r
4 r

 R nl
 
2
. To find probability over a whole range of r, integrate with respect to r.
   lm

General form of hydrogen wavefunctions. R is the radial wavefunction and Y is the spherical harmonic. They are independent of each other.
L 2    1

2
Very important. L is total angular momentum. l is the familiar quantum number.
L z
 m
Also important. Angular momentum in z direction is proportional to m quantum number.
, , Y
10
= …
The spherical harmonics for l =0 and l =1.
U    B
Potential energy of a particle in a magnetic field is equal to magnetic moment times field strength.
F z
   z dB z dz
: Follows directly from above. Take derivate w.r.t z. e
   z
S z Magnetic moment of electron. e is electric charge. m e
is mass. S z
is the spin of the electron m e
E n in the z direction.
 n
 
1
2

: energy levels of the harmonic oscillator.
S 2    1

2
: S is the spin angular momentum. s is the spin quantum number.
S z
 m s
: S z
is the z component of the spin angular momentum. m s
is another quantum number related to spin s , just like m l
relates to l .